Direct product of groups: Let (G, ∗G) and (H, ∗H) be groups, with identity elements eG...

Direct product of groups: Let (G, ∗_G) and (H, ∗_H) be groups, with identity elements e_G and e_H, respectively. Let g be any element of G, and h any element of H. (a) Show that the set G × H has a natural group structure under the operation (∗_G, ∗_H). What is the identity element of G × H with this structure? What is the inverse of the element (g, h) ∈ G × H? (b) Show that the map i_G : G → G × H given by i_G(g) = (g, e_H) is a group homomorphism. Is it injective? Surjective? Do the same for the map i_H : H → G × H given by i_H(h) = (e_G, h). (c) Show that the map π_G : G × H → G given by π_G (g, h) = g is a group homomorphism. Is it injective? Surjective? Do the same for the map π_H : G × H → H given by π_H (g, h) = h. (d) Prove that the image of i_G is the kernel of π_H, and that the image of i_H is the kernel of π_G

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Colby Messinger answered 1 year ago

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